Fixed Parameter Complexity Algorithms and Networks

2 Fixed parameter complexity Analysis what happens to problem when some parameter is small Today: –Definitions –Fixed parameter tractability techniques Branching Kernelisation Other techniques

3 Motivation In many applications, some number may be assumed to be small –Time of algorithm can be exponential in this small number, but should be polynomial in usual size of problem

Example: Can you pick you pick 7 vertices that hit all edges? 4

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6 Parameterized problem Given: Graph G, integer k, … Parameter: k Question: Does G have a ??? of size at least (at most) k? –Examples: vertex cover, independent set, coloring, …

7 Examples of parameterized problems (1) Graph Coloring Given: Graph G, integer k Parameter: k Question: Is there a vertex coloring of G with k colors? (I.e., c: V  {1, 2, …, k} with for all {v,w}  E: c(v)  c(w)?) NP-complete, even when k=3.

8 Examples of parameterized problems (2) Clique Given: Graph G, integer k Parameter: k Question: Is there a clique in G of size at least k? Solvable in O(n k+1 ) time with simple algorithm. Complicated algorithm gives O(n 2.23k/3 ). Seems to require  (n f(k) ) time…

9 Examples of parameterized problems (3) Vertex cover Given: Graph G, integer k Parameter: k Question: Is there a vertex cover of G of size at most k? Solvable in O(2 k (n+m)) time

10 Fixed parameter complexity theory To distinguish between behavior:  O( f(k) * n c )   ( n f(k) ) Proposed by Downey and Fellows.

11 Parameterized problems Instances of the form (x,k) –I.e., we have a second parameter Decision problem (subset of {0,1}* x N )

12 Fixed parameter tractable problems FPT is the class of problems with an algorithm that solves instances of the form (x,k) in time p(|x|)*f(k), for polynomial p and some function f.

13 Hard problems Complexity classes –FPT  W[1]  W[2]  … W[i]  …  W[P] –FPT is ‘easy’, all others ‘hard’ –Defined in terms of Boolean circuits –Problems hard for W[1] or larger classes are assumed not to be in FPT Compare with P / NP

14 Examples of hard problems Clique and Independent Set are W[1]-complete Dominating Set is W[2]-complete This version of Satisfiability is W[1]-complete –Given: set of clauses, k –Parameter: k –Question: can we set (at most) k variables to true, and al others to false, and make all clauses true?

15 Techniques for showing fixed parameter tractability Branching Kernelisation Iterative compression Other techniques (e.g., treewidth)

16 A branching algorithm for vertex cover Idea: –Simple base cases –Branch on an edge: one of the endpoints belongs to the vertex cover Input: graph G and integer k

17 Recursive procedure VC(Graph G, int k) VC(G=(V,E), k) –If G has no edges, then return true –If k == 0, then return false –Select an edge {v,w}  E –Compute G’ = G [V – v] –Compute G” = G [V – w] –Return VC(G’,k – 1) or VC(G”,k – 1) A branching algorithm for vertex cover

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24 Analysis of algorithm Correctness –Either v or w must belong to an optimal VC Time analysis –Branching tree has 2 k leaves, so 2 k recursive calls –Each recursive call costs O(n+m) time –O(2 k (n+m)) time: FPT

25 Cluster editing Instance: undirected graph G=(V,E), integer K Parameter: K Question: can we make at most K modifications to G, such that each connected component is a clique, where each modification is an addition of an edge or the deletion of an edge? Models biological question: partition species in families, where available data contains mistakes NP-complete. Branching: O(3 k p(n)) algorithm

26 Lemma If G has a connected component that is not a clique, then G contains the following subgraph: Proof: there are vertices w and x in the connected component that are not adjacent. Take such w and x of minimum distance, which must be 2. w vx

27 Branching algorithm for Cluster Editing If each connected component is a clique: –Answer YES If k=0 and some connected components are not cliques: –Answer NO Otherwise, there must be vertices v, w, x with {v,w}  E, {v,x}  E, and {w,x}  E –Go three times in recursion: Once with {v,w} removed and k = k – 1 Once with {v,x} removed and k = k – 1 Once with {w,x} added and k = k – 1 w vx

28 Analysis branching algorithm Correctness by lemma Time analysis: branching tree has 3 k leaves

Fixed Parameter Complexity29 More on cluster editing Faster branching algorithms exist Important applications and practical experiments We’ll see more when discussing kernelisation

Fixed Parameter Complexity30 Max SAT Variant of satisfiability, but now we ask: can we satisfy at least k clauses? NP-complete With k as parameter: FPT Branching: –Take a variable –If it only appears positively, or negatively, then … –Otherwise: Branch! What happens with k?

Independent Set on Planar Graphs Given: a planar graph G=(V,E), integer k Parameter: k Question: Does G have an independent set of at least k vertices, i.e., a set W of size at least k, such that for all v, w in W: {v,w} is not an edge? NP-complete problem Here we argue O(6 k n) algorithm (faster is possible) Fixed Parameter Complexity31

32 The red vertices form an independent set

33 Branching Each planar graph has a vertex of degree at most 5 Take vertex v of minimum degree, say with neighbors w 1, …, w r, r at most 5 A maximum size independent set contains v or one of its neighbors –Selecting a vertex is equivalent to removing it and its neighbors and decreasing k by one Create at most 6 subproblems, one for each x  {v, w 1, …, w r }. In each, we set k = k – 1, and remove x and its neighbors

34 v w2w2 w1w1

35 Closest string Given: k strings s 1, …,s k each of length L, integer d Parameter: d Question: is there a string s with Hamming distance at most d to each of s 1, …,s k Application in molecular biology Here: FPT algorithm (Gramm and Niedermeier, 2002)

36 Subproblems Subproblems have form –Candidate string s –Additional parameter r –We look for a solution to original problem, with additional condition: Hamming distance at most r to s Start with s = s 1 and r=d (= original problem)

37 Branching step Choose an s j with Hamming distance > d to s –If none exists, s is a solution If Hamming distance of s j to s > d+r: answer NO For all positions i where s j differs from s –Solve subproblem with s changed at position i to value s j (i) r = r – 1 Note: we find a solution, if and only one of these subproblems has a solution

38 Example Strings 01113, 02223, 01221, d=2 gives –(02113, 1) (02213, 0)  (02123, 0)  –(01213, 1) (02213, 0)  (01223, 0) –(01123, 1) (02123, 0)  (01223, 0)

39 Time analysis Recursion depth d At each level, we branch at most at d + r  2d positions So, number of recursive steps at most (2d) d+1 Each step can be done in polynomial time: O(kdL) Total time is O((2d) d+1. kdL) Speed up possible by more clever branching and by kernelisation

40 More clever branching step Choose an s j with Hamming distance > d to s –If none exists, s is a solution If Hamming distance of s j to s > d+r: answer NO Choose pick d+1 positions where s j differs from s –Solve subproblem with s changed at position i to value s j (i) r = r – 1 Still correct since the solution differs with s on at least one of these positions. O(kL + kd (d+1) d ) time

41 Technique Try to find a branching rule that –Decreases the parameter –Splits in a bounded number of subcases YES, if and only if YES in at least one subcase

Kernelization

43 Kernelisation Preprocessing rules reduce starting instance to one of size f(k) –Should work in polynomial time Then use any algorithm to solve problem on kernel Time will be p(n) + g(f(k))

44 Kernelisation Helps to analyze preprocessing Much recent research Today: definition and some examples

45 Formal definition of kernelisation Let P be a parameterized problem. (Each input of the form (I,k).) A reduction to a problem kernel is an algorithm A, that transforms inputs of P to inputs of P, such that –(I,k)  P, if and only if A(I,k)  P for all (I,k) –If A(I,k) = (I’,k’), then k’  f(k), and |I’|  g(k) for some functions f, g –A uses time, polynomial in |I| and k

Fixed Parameter Complexity46 Kernels and FPT Theorem. Consider a decidable parameterized problem. Then the problem belongs to FPT, if and only if it has a kernel <= Build kernel and solve the problem on kernel => Suppose we have an f(k)n c algorithm. Run the algorithm for n c+1 steps. If it did not yet solve the problem, return the input as kernel: it has size at most f(k). If it solved the problem, then return small YES / NO instance

Fixed Parameter Complexity47 Consequence If a problem is W[1]-hard, it has no kernel, unless FPT=W[1] There also exist techniques to give evidence that problems have no kernels of polynomial size –If problem is compositional and NP-hard, then it has no polynomial kernel –Example is e.g., LONG PATH

First kernel: Convex string recoloring Application from molecular biology; NP-complete Convex String Recoloring –Given: string s in  *, integer k –Parameter: k –Question: can we change at most k characters in the string s, such that s becomes convex, i.e., for each symbol, the positions with that symbol are consecutive. Example of convex string: aaacccbxxxffff Example of string that is not convex: abba Instead of symbols, we talk about colors 48

Kernel for convex string recoloring Theorem: Convex string recoloring has a kernel with O(k 2 ) characters. 49

Notions Notion: good and bad colors A color is good, if it is consecutive in s, otherwise it is bad abba: a is bad and b is good Notion of block: consecutive occurrences of the same color, ie. aabbacc has four blocks Convex: each color has one block 50

Construction of kernel Apply three reduction rules –Rule 1: limit #blocks of bad colors –Rule 2: limit #different good colors –Rule 3: limit #characters in s per block Count 51

Rule 1 If there are more than 4k blocks of bad colors, say NO –Formally, transform to trivial NO-instance, e.g. (aba, 0) –Why correct? Each change can decrease the number of blocks of bad colors by at most 4. –Example: abab -> abbb 52

Rule 2 If we have two consecutive blocks of good colors, then change the color of the second block to that of the first E.g: abbbbcca -> abbbbbba Why correct? –We will only recolor the c’s to connect bad colors, and this still comes at the same cost. 53

Rule 3 If a block has more than k+1 characters, delete all but k+1 of the block Correctness: a block of such a size can never be changed 54

Counting After the rules have been applied, we have at most: –4k blocks of bad colors –4k+1 blocks of good colors: at most one between each pair of bad colors, one in front and one in the end –Each block has size at most k+1 String has size at most (8k+1)(k+1) This can be improved by better analysis, more rules,... 55

Fixed Parameter Complexity (2 nd half) Reminder Kernelization: –Point line cover (warm-up) –Vertex Cover –Max Sat –Cluster editing –Non-blocker (might skip it) FPT techniques: –Iterative Compression –Color Coding Fixed Parameter Complexity56

57 Reminder: Fixed Parameter Complexity Given: Graph G, integer k, … Parameter: k Question: Does G have a ??? of size at least (at most) k? Distinguish between behavior:  O( f(k) * n c )   ( n f(k) )

58 Reminder: Kernelisation Let P be a parameterized problem. (Each input of the form (I,k).) A reduction to a problem kernel is an algorithm A, that transforms inputs of P to inputs of P, such that –(I,k)  P, if and only if A(I,k)  P for all (I,k) –If A(I,k) = (I’,k’), then k’  f(k), and |I’|  g(k) for some functions f, g –A uses time, polynomial in |I| and k

Fixed Parameter Complexity59 Kernel: Point-line cover In: n points in the plane, integer k Question: Can you hit all the points with k straight lines? NP-Complete.

Quadratic kernel Rule 1: If some line covers k+1 points use it (and reduce k by one). Rule 2: If Rule 1 is not applicable and n > k 2, say NO. Kernel with k 2 points! 60

61 Vertex cover: observations that help for kernelisation Isolated vertices can be ignored If v has degree at least k+1, v belongs to each vertex cover in G of size at most k. –If v is not in the vertex cover, then all its neighbors are in the vertex cover. If all vertices have degree at most k, and m > k 2, there is no vertex cover of size k.

62 Kernelisation for Vertex Cover H = G; ( S =  ; ) While there is a vertex v in H of degree at least k+1 do Remove v and its incident edges from H k = k – 1; ( S = S + v ;) If k < 0 then return false If H has at least k 2 +1 edges, then return false Remove vertices of degree 0

63 Time Kernelisation step can be done in O(n+m) time After kernelisation, we must solve the problem on a graph with at most k 2 edges, e.g., with branching this gives: –O( n + m + 2 k k 2 ) time –O( kn + 2 k k 2 ) time can be obtained by noting that there is no solution when m > kn.

64 Better kernel for vertex cover Nemhauser-Trotter: kernel of at most 2k vertices Make ILP formulation of Vertex Cover Solve relaxation All vertices v with x v > ½ : put v in set All vertices v with x v < ½ : v is not in the set Remove all vertices except those with value ½, and decrease k accordingly Gives kernel with at most 2k vertices, but why is it correct?

65 Nemhauser Trotter proof plan

66 1. ILP

67 2. Relaxation

3. Observation Fixed Parameter Complexity68

3. Observation Fixed Parameter Complexity69 added deleted

3. Observation Fixed Parameter Complexity70 added deleted

71 2k kernel for Vertex Cover Solve the relaxation (polynomial time with the ellipsoid method, practical with Simplex) If the relaxation has optimum more than k, then say no Otherwise, get rid of the 0’s and 1’s, decrease k accordingly At most 2k vertices have weight ½ in the relaxation So, kernel has 2k vertices. It can (and will often) have a quadratic number of edges

72 Maximum Satisfiability Given: Boolean formula in conjunctive normal form; integer k Parameter: k Question: Is there a truth assignment that satisfies at least k clauses? Denote number of clauses with C

73 Reducing the number of clauses

74 Bounding number of long clauses Long clause: has at least k literals Short clause: has at most k-1 literals Let L be number of long clauses If L  k: answer is YES –Select in each of the first k long clauses a literal, whose complement is not yet selected –Set these all to true –At least k long clauses are satisfied

75 Reducing to only short clauses If less than k long clauses –Make new instance, with only the short clauses and the parameter set to k-L –Valid, because: there is a truth assignment that satisfies at least k-L short clauses, if and only if there is a truth assignment that satisfies at least k clauses =>: choose for each satisfied short clause a variable that makes the clause true. We may change all other variables, and can choose for each long clause another variable that makes it true <=: there are only L long clause so we satisfy k-L short ones

76 An O(k 2 ) kernel for Maximum Satisfiability If at least 2k clauses then return YES If at least k long clauses then return YES Else –remove all L long clauses –set k=k-L

77 Reminder: Cluster editing Instance: undirected graph G=(V,E), integer K Parameter: K Question: can we make at most K modifications to G, such that each connected component is a clique, where each modification is an addition of an edge or the deletion of an edge?

78 Kernelisation for cluster editing Quadratic kernel General form: Repeat rules, until no rule is possible –Rules can do some necessary modification and decrease k by one –Rules can remove some part of the graph –Rules can output YES or NO

79 Trivial rules and plan Rule 1: If a connected component of G is a clique, remove this connected component Rule 2: If we have more than k connected components and Rule 1 does not apply: Answer NO Consequence: after Rule 1 and Rule 2, there are at most k connected components Plan: find rules that make connected component small We change the input: some pairs are permanent and others are forbidden.

80 Observation and rule 3 If two vertices v, w have k+1 neighbors in common, they must belong to the same clique in a solution Rule 3: If v, w have k+1 neighbors in common, then –If {v,w} did not exist, add it, and set k = k – 1. –Set the edge {v,w} to be permanent

81 Another observation and rule 4 If there are at least k+1 vertices that are adjacent to exactly one of v and w, then {v,w} cannot be an edge in the solution Rule 4: If there are at least k+1 vertices that are adjacent to exactly one of v and w, then –If {v,w} is an edge: delete it and decrease k by one –Mark the pair {v,w} as forbidden v w

A trivial rule Rule 5: if a pair is forbidden and permanent then there is no solution Fixed Parameter Complexity82

83 Transitivity Rule 6: if {v,w} is permanent, and {w,x} is permanent, then set {v,x} to be permanent (if the edge was nonexisting, add it, and decrease k by one) Rule 7: if {v,w} is permanent and {w,x} is forbidden, then set {v,x} to be forbidden (if the edge existed, delete it, and decrease k by one)

84 Counting Rules can be executed in polynomial time One can find in O(n 3 ) time an instance to which no rules apply (with properly chosen data structures) Claim: If no rule applies, and the instance is a YES-instance, then every connected component has size at most 4k. Proof: Suppose a YES-instance, and consider a connected component C of size at least 4k+1. At least 2k+1 vertices are not involved in a modification, say this is the set W W must form a clique, and all edges in W become permanent (Rule 3) …

85 Counting continued

Conclusion Rule 8: If no other rule applies, and there is a connected component with at least 4k+1, vertices, say NO. As we have at most k connected components, each of size at most 4k, our kernel has at most 4k 2. Fixed Parameter Complexity86

87 Comments This argument is due to Gramm et al. Better and more recent algorithms exist: faster algorithm (2.7 k ) and linear kernels

88 Non-blocker Given: graph G=(V,E), integer k Parameter k Question: Does G have a dominating set of size at most |V|-k ?

89 Lemma and simple linear kernel If G does not have vertices of degree 0, then G has a dominating set with at most |V|/2 vertices –Proof: per connected component: build spanning tree. The vertices on the odd levels form a ds, and the vertices on the even levels form a ds. Take the smallest of these. 2k kernel for non-blocker after removing vertices of degree 0

90 Improvements Lemma (Blank and McCuaig, 1973) If a connected graph has minimum degree at least two and at least 8 vertices, then the size of a minimum dominating set is at most 2|V|/5. Lemma (Reed) If a connected graph has minimum degree at least three, then the size of a minimum dominating set is at most 3|V|/8. Can be used by applying reduction rules for killing degree one and degree two vertices

91 Iterative compression

92 Feedback Vertex Set Instance: graph G=(V,E) Parameter: integer k Question: Is there a set of at most k vertices W, such that G-W is a forest? –Known in FPT –Here: recent algorithm O(5 k p(n)) time algorithm –Can be done in O(5 k kn) or less with kernelisation –Fastest known algorithm (randomized): O(3 k p(n)), invented at Utrecht, using treewidth techniques.

93 Iterative compression technique Number vertices v 1, v 2, …, v n Let X = {v 1, v 2, …, v k } for i = k+1 to n do Add v i to X –Note: X is a FVS of size at most k+1 of G[{v 1, v 2, …, v i }] Call a subroutine that either –Finds (with help of X) a feedback vertex set Y of size at most k in {v 1, v 2, …, v i }; set X = Y OR –Determines that such Y does not exist; stop, return NO

94 Compression subroutine Given: graph G, FVS X of size k + 1 Question: find if existing FVS of size k –Is subroutine of main algorithm  for all subsets S of X do Determine if there is a FVS of size at most k that contains all vertices in S and no vertex in X – S

95 Yet a deeper subroutine Given: Graph G, FVS X of size k+1, set S Question: find if existing a FVS of size k containing all vertices in S and no vertex from X – S 1.Remove all vertices in S from G 2.Mark all vertices in X – S 3.If marked cycles contain a cycle, then return NO 4.While marked vertices are adjacent, contract them 5.Set k = k - |S|. If k < 0, then return NO 6.If G is a forest, then return YES; S 7.…

G[{v 1, v 2, …, v i }] X Fixed Parameter Complexity96 Compression subroutine S S Answer NO, we cannot kill this cycle since we don’t remove any vertex.

G[{v 1, v 2, …, v i }] X Fixed Parameter Complexity97 Compression subroutine

98 Subroutine continued 7.If an unmarked vertex v has at least two edges to marked vertices If these edges are parallel, i.e., to the same neighbor, then v must be in a FVS (we have a cycle with v the only unmarked vertex) Put v in S, set k = k – 1 and recurse Else recurse twice: Put v in S, set k = k – 1 and recurse Mark v, contract v with all marked neighbors and recurse –The number of marked vertices is one smaller

G[{v 1, v 2, …, v i }] X Fixed Parameter Complexity99 Compression subroutine

G[{v 1, v 2, …, v i }] X Fixed Parameter Complexity100 Compression subroutine

G[{v 1, v 2, …, v i }] X Fixed Parameter Complexity101 Compression subroutine

102 Other case All unmarked vertices have at most 2 marked neighbors! 8.Choose an unmarked vertex v that has at most one unmarked neighbor (a leaf in G[V-X])  By step 7, it also has at most one marked neighbor If v is a leaf in G, then remove v If v has degree 2, then remove v and connect its neighbors

G[{v 1, v 2, …, v i }] X Fixed Parameter Complexity103 Other case

104 Analysis Precise analysis gives O*(5 k ) subproblems in total Imprecise: 2 k subsets S Only branching step: –k is decreased by one, or –Number of marked vertices is decreased by one Initially: number of marked vertices + k is at most 2k Bounded by 2 k.2 2k = 8 k

Parameterized Complexity105 Color coding Interesting algorithmic technique to give fast FPT algorithms As example: Long Path –Given: Graph G=(V,E), integer k –Parameter: k –Question: is there a simple path in G with at least k vertices?

Long Path on Directed Acyclic Graphs (DAG) Consider long path on DAG’s. Polynomial time easily! Fixed Parameter Complexity106

Parameterized Complexity107 Problem on colored graphs Given: graph G=(V,E), for each vertex v a color in {1,2, …, k} Question: Is there a simple path in G with k vertices of different colors? –Note: vertices with the same colors may be adjacent –Can be solved in O(k! (n+m)) (try all orders of colors and reduce to long paths in DAG) –Better: O(2 k (nm)) time using dynamic programming Used as subroutine…

Parameterized Complexity108 DP Tabulate: –(S,v): S is a set of colors, v a vertex, such that there is a path using vertices with colors in S, and ending in v –Using Dynamic Programming, we can tabulate all such pairs, and thus decide if the requested path exists

Parameterized Complexity109 A randomized variant

110 Conclusions Similar techniques work (usually much more complicated) for many other problems W[…]-hardness results indicate that FPT- algorithms do not exist for other problems Note similarities and differences with exponential time algorithms